Qual2K: a modeling Framework for Simulating River and Stream Water Quality (Version 11)
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Constituent ReactionsThe mathematical relationships that describe the individual reactions and concentrations of the model state variables (Table ) are presented in the following paragraphs.
By definition, conservative substances are not subject to reactions: Ss = 0 ()
Phytoplankton increase due to photosynthesis. They are lost via respiration, death, and settling  ()
Phytoplankton photosynthesis is computed as  ()
where ?p = phytoplankton photosynthesis rate [/d] is a function of temperature, nutrients, and light,  ()
where kgp(T) = the maximum photosynthesis rate at temperature T [/d], ?Np = phytoplankton nutrient attenuation factor [dimensionless number between 0 and 1], and ?Lp = the phytoplankton light attenuation coefficient [dimensionless number between 0 and 1]. Nutrient Limitation. The nutrient limitation due to inorganic carbon is represented by a Michaelis-Menten. In contrast, for nitrogen and phosphorus, the photosynthesis rate depends on intracellular nutrient levels using a formulation originally developed by Droop (1974). The minimum value is then employed to compute the nutrient attenuation factor,  ()
where qNp and qPp = the phytoplankton cell quotas of nitrogen [mgN mgA?1] and phosphorus [mgP mgA?1], respectively, q0Np and q0Pp = the minimum phytoplankton cell quotas of nitrogen [mgN mgA?1] and phosphorus [mgP mgA?1], respectively, ksCp = inorganic carbon half-saturation constant for phytoplankton [mole/L], [H2CO3*] = dissolved carbon dioxide concentration [mole/L], and [HCO3?] = bicarbonate concentration [mole/L]. The minimum cell quotas are the levels of intracellular nutrient at which growth ceases. Note that the nutrient limitation terms cannot be negative. That is, if q < q0, the limitation term is set to 0. The cell quotas represent the ratios of the intracellular nutrient to the phytoplankton biomass,  ()
 ()
where INp = phytoplankton intracellular nitrogen concentration [?gN/L] and IPp = phytoplankton intracellular phosphorus concentration [?gP/L]. Light Limitation. It is assumed that light attenuation through the water follows the Beer-Lambert law,  ()
where PAR(z) = photosynthetically available radiation (PAR) at depth z below the water surface [ly/d]3, and ke = the light extinction coefficient [m?1]. The PAR at the water surface is assumed to be a fixed fraction of the solar radiation (Szeicz 1984, Baker and Frouin 1987): PAR(0) = 0.47 I(0) The extinction coefficient is related to model variables by  ()
where keb = the background coefficient accounting for extinction due to water and color [/m], ?i, ?o, ?p, and ?pn, are constants accounting for the impacts of inorganic suspended solids [L/mgD/m], particulate organic matter [L/mgD/m], and chlorophyll [L/?gA/m and (L/?gA)2/3/m], respectively. Suggested values for these coefficients are listed in Table . Table Suggested values for light extinction coefficients
Three models are used to characterize the impact of light on phytoplankton photosynthesis (Figure ): 
Figure The three models used for phytoplankton and bottom algae photosynthetic light dependence. The plot shows growth attenuation versus PAR intensity [ly/d]. Half-Saturation (Michaelis-Menten) Light Model (Baly 1935):  ()
where FLp = phytoplankton growth attenuation due to light and KLp = the phytoplankton light parameter. In the case of the half-saturation model, the light parameter is a half-saturation coefficient [ly/d]. This function can be combined with the Beer-Lambert law and integrated over water depth, H [m], to yield the phytoplankton light attenuation coefficient  ()
Smith’s Function (Smith 1936):  ()
where KLp = the Smith parameter for phytoplankton [ly/d]; that is, the PAR at which growth is 70.7% of the maximum. This function can be combined with the Beer-Lambert law and integrated over water depth to yield  ()
Steele’s Equation (Steele 1962):  ()
where KLp = the PAR at which phytoplankton growth is optimal [ly/d]. This function can be combined with the Beer-Lambert law and integrated over water depth to yield  ()
Respiration. Phytoplankton respiration is represented as a first-order rate that is attenuated at low oxygen concentration,  ()
where krp(T) = temperature-dependent phytoplankton respiration/excretion rate [/d] and Foxp = attenuation due to low oxygen [dimensionless]. Oxygen attenuation is modeled by Eqs. (127) to (129) with the oxygen dependency represented by the parameter Ksop. Death. Phytoplankton death is represented as a first-order rate,  ()
where kdp(T) = temperature-dependent phytoplankton death rate [/d]. Settling. Phytoplankton settling is represented as  ()
where va = phytoplankton settling velocity [m/d].
The change in intracellular nitrogen in phytoplankton cells is calculated from  ()
where PhytoUpN = the uptake rate of nitrogen by phytoplankton (?gN/L/d), PhytoDeath = phytoplankton death (?gN/L/d), and PhytoExN = the phytoplankton excretion of nitrogen (?gN/L/d), which is computed as  ()
where kep(T) = the temperature-dependent phytoplankton excretion rate [/d]. The N uptake rate depends on both external and intracellular nutrients as in (Rhee 1973),  ()
where ?mNp = the maximum uptake rate for nitrogen [mgN/mgA/d], ksNp = half-saturation constant for external nitrogen [?gN/L] and KqNp = half-saturation constant for intracellular nitrogen [mgN mgA?1].
The change in intracellular phosphorus in phytoplankton cells is calculated from  ()
where PhytoUpP = the uptake rate of phosphorus by phytoplankton (?gP/L/d), PhytoDeath = phytoplankton death (?gP/L/d), and PhytoExP = the phytoplankton excretion of phosphorus (?gP/L/d), which is computed as  ()
where kep(T) = the temperature-dependent phytoplankton excretion rate [/d]. The P uptake rate depends on both external and intracellular nutrients as in (Rhee 1973),  ()
where ?mPp = the maximum uptake rate for phosphorus [mgP/mgA/d], ksPp = half-saturation constant for external phosphorus [?gP/L] and KqPp = half-saturation constant for intracellular phosphorus [mgP mgA?1].
Bottom algae increase due to photosynthesis. They are lost via respiration and death.  ()
Two representations can be used to model bottom algae photosynthesis. The first is based on a temperature-corrected zero-order rate attenuated by nutrient and light limitation (McIntyre 1973, Rutherford et al. 1999),  ()
where Cgb(T) = the zero-order temperature-dependent maximum photosynthesis rate [mgA/(m2 d)], ?Nb = bottom algae nutrient attenuation factor [dimensionless number between 0 and 1], and ?Lb = the bottom algae light attenuation coefficient [dimensionless number between 0 and 1]. The second uses a first-order model,  ()
where, for this case, Cgb(T) = the first-order temperature-dependent maximum photosynthesis rate [d?1], and ?Sb = bottom algae space limitation attenuation factor. Temperature Effect. As for the first-order rates, an Arrhenius model is employed to quantify the effect of temperature on bottom algae photosynthesis,  ()
Nutrient Limitation. The effect of nutrient limitation on bottom plant photosynthesis is modeled in the same way as for the phytoplankton. That is, a Droop (1974) formulation is used for nitrogen and phosphorus limitation whereas a Michaelis-Menten equation is employed for inorganic carbon,  ()
where qNb and qPb = the bottom algae cell quotas of nitrogen [mgN mgA?1] and phosphorus [mgP mgA?1], respectively, q0Nb and q0Pb = the bottom algae minimum cell quotas of nitrogen [mgN mgA?1] and phosphorus [mgP mgA?1], respectively, and , ksCb = the bottom algae inorganic carbon half-saturation constant [mole/L]. As was the case for phytoplankton, the nutrient limitation terms cannot be negative. The cell quotas represent the ratios of the intracellular nutrient to the bottom plant’s biomass,  ()
 ()
where INb = intracellular nitrogen concentration [mgN/m2] and IPb = intracellular phosphorus concentration [mgP/m2]. Light Limitation. In contrast to the phytoplankton, light limitation at any time is determined by the amount of PAR reaching the bottom of the water column. This quantity is computed with the Beer-Lambert law (recall Eq. 83) evaluated at the bottom of the river,  ()
As with the phytoplankton, three models (Eqs. 85, 87 and 89) are used to characterize the impact of light on bottom algae photosynthesis. Substituting Eq. (107) into these models yields the following formulas for the bottom algae light attenuation coefficient, Half-Saturation Light Model (Baly 1935):  ()
Smith’s Function: (Smith 1936)  ()
Steele’s Equation (Steele 1962):  ()
where KLb = the appropriate bottom algae light parameter for each light model. Space Limitation. If a first-order growth model is used, a term must be included to impose a space limitation on the bottom algae. A logistic model is used for this purpose as in 
where ab,max = the carrying capacity [mgA/m2].
Respiration. Bottom algae respiration is represented as a first-order rate that is attenuated at low oxygen concentration,  ()
where krb(T) = temperature-dependent bottom algae respiration rate [/d] and Foxb = attenuation due to low oxygen [dimensionless]. Oxygen attenuation is modeled by Eqs. (127) to (129) with the oxygen dependency represented by the parameter Ksob. Death. Bottom algae death is represented as a first-order rate,  ()
where kdb(T) = the temperature-dependent bottom algae death rate [/d].
The change in intracellular nitrogen in bottom algal cells is calculated from  ()
where BotAlgUpN = the uptake rate of nitrogen by bottom algae (mgN/m2/d), BotAlgDeath = bottom algae death (mgA/m2/d), and BotAlgExN = the bottom algae excretion of nitrogen (mgN/m2/d), which is computed as  ()
where keb(T) = the temperature-dependent bottom algae excretion rate [/d]. The N uptake rate depends on both external and intracellular nutrients as in (Rhee 1973),  ()
where ?mNb = the maximum uptake rate for nitrogen [mgN/mgA/d], ksNb = half-saturation constant for external nitrogen [?gN/L] and KqNb = half-saturation constant for intracellular nitrogen [mgN mgA?1].
The change in intracellular phosphorus in bottom algal cells is calculated from  ()
where BotAlgUpP = the uptake rate of phosphorus by bottom algae (mgP/m2/d), BotAlgDeath = bottom algae death (mgA/m2/d), and BotAlgExP = the bottom algae excretion of phosphorus (mgP/m2/d), which is computed as  ()
where keb(T) = the temperature-dependent bottom algae excretion rate [/d]. The P uptake rate depends on both external and intracellular nutrients as in (Rhee 1973),  ()
where ?mPb = the maximum uptake rate for phosphorus [mgP/mgA/d], ksPb = half-saturation constant for external phosphorus [?gP/L] and KqPb = half-saturation constant for intracellular phosphorus [mgP mgA?1].
Detritus or particulate organic matter (POM) increases due to plant death. It is lost via dissolution and settling  ()
where  ()
where kdt(T) = the temperature-dependent detritus dissolution rate [/d] and  ()
where vdt = detritus settling velocity [m/d].
Slowly reacting CBOD increases due to detritus dissolution. It is lost via hydrolysis and oxidation,  ()
where Ff = the fraction of detrital dissolution that goes to fast reacting CBOD [dimensionless], and  ()
where khc(T) = the temperature-dependent slow CBOD hydrolysis rate [/d], and  ()
where kdcs(T) = the temperature-dependent slow CBOD oxidation rate [/d] and Foxc = attenuation due to low oxygen [dimensionless].
Fast reacting CBOD is gained via the dissolution of detritus and the hydrolysis of slowly-reacting CBOD. It is lost via oxidation and denitrification.  ()
where  ()
where kdc(T) = the temperature-dependent fast CBOD oxidation rate [/d] and Foxc = attenuation due to low oxygen [dimensionless]. The parameter rondn is the ratio of oxygen equivalents lost per nitrate nitrogen that is denitrified (Eq. 67). The term Denitr is the rate of denitrification [?gN/L/d]. It will be defined in Sec. 5.5.15 below. Three formulations are used to represent the oxygen attenuation: Half-Saturation:  ()
where Ksocf = half-saturation constant for the effect of oxygen on fast CBOD oxidation [mgO2/L]. Exponential:  ()
where Ksocf = exponential coefficient for the effect of oxygen on fast CBOD oxidation [L/mgO2]. Second-Order Half Saturation:  ()
where Ksocf = half-saturation constant for second-order effect of oxygen on fast CBOD oxidation [mgO22/L2].
Organic nitrogen increases due to plant death. It is lost via hydrolysis and settling.  ()
where fonp = the fraction of the phytoplankton internal nitrogen that is in organic form which is calculated as  ()
The fraction of the bottom algae internal nitrogen that is in organic form, fonb, is calculated in a similar fashion. The rate of organic nitrogen hydrolysis is computed as  ()
where khn(T) = the temperature-dependent organic nitrogen hydrolysis rate [/d]. Organic nitrogen settling is determined as  ()
where von = organic nitrogen settling velocity [m/d].
Ammonia nitrogen increases due to organic nitrogen hydrolysis and plant death and excretion. It is lost via nitrification and plant photosynthesis:  ()
The ammonia nitrification rate is computed as  ()
where kn(T) = the temperature-dependent nitrification rate for ammonia nitrogen [/d] and Foxna = attenuation due to low oxygen [dimensionless]. Oxygen attenuation is modeled by Eqs. (127) to (129) with the oxygen dependency represented by the parameter Ksona. The coefficients Pap and Pab are the preferences for ammonium as a nitrogen source for phytoplankton and bottom algae, respectively,  ()
 ()
where khnxp = preference coefficient of phytoplankton for ammonium [mgN/m3] and khnxb = preference coefficient of bottom algae for ammonium [mgN/m3].
The model simulates total ammonia. In water, the total ammonia consists of two forms: ammonium ion, NH4+, and unionized ammonia, NH3. At normal pH (6 to 8), most of the total ammonia will be in the ionic form. However at high pH, unionized ammonia predominates. The amount of unionized ammonia can be computed as  ()
where nau = the concentration of unionized ammonia [?gN/L], and Fu = the fraction of the total ammonia that is in unionized form,  ()
where Ka = the equilibrium coefficient for the ammonia dissociation reaction, which is related to temperature by  ()
where Ta is absolute temperature [K] and pKa = ?log10(Ka). Note that the fraction of the total ammonia that is in ionized form, Fi, can be computed as 1 – Fu or  ()
The loss of ammonia via gas transfer is computed as 
where vnh3(T) = the temperature-dependent ammonia gas-transfer coefficient [m/d], and nas(T) = the saturation concentration of ammonia [?gN/L] at temperature, T. The transfer coefficient is calculated as 
where vv = the mass-transfer coefficient (m/d), Kl and Kg = liquid and gas film exchange coefficients [m/d], respectively, R = the universal gas constant (= 8.206?10–5 atm m3/(K mole)), Ta = absolute temperature [K], and He = Henry’s constant (atm m3/mole). The saturation concentration is calculated as 
where pnh3 = the partial pressure of ammonia in the atmosphere (atm), and CF is a conversion factor (?gN/L per moleNH3/m3). The partial pressure of ammonia ranges from 1-10 ppb in rural and moderately polluted areas to 10-100 ppb in heavily polluted areas (Holland 1978, Finlayson-Pitts and Pitts 1986). We will assume that a value of 2 ppb, which corresponds to 2?10–9 atm, represents a typical value. The conversion factor is 
The liquid-film coefficient can be related to the oxygen reaeration rate by (Mills et al. 1982), 
The gas-film coefficient is computed by 
Kg,H2O = a mass transfer velocity for water vapor [m/d], which can be related to wind speed by (Schwarzenbach et al. 1993) 
where Uw,10 = wind speed at a height of 10 m (m/s). Combining the above equations gives 
The Henry’s constant for ammonia at 20oC is 1.3678?10–5 atm m3/mole (Kavanaugh and Trussell 1980). The value at temperatures other than 20oC can be computed with 
Nitrate nitrogen increases due to nitrification of ammonia. It is lost via denitrification and plant uptake:  ()
The denitrification rate is computed as  ()
where kdn(T) = the temperature-dependent denitrification rate of nitrate nitrogen [/d] and Foxdn = effect of low oxygen on denitrification [dimensionless] as modeled by Eqs. (127) to (129) with the oxygen dependency represented by the parameter Ksodn.
Organic phosphorus increases due to plant death and excretion. It is lost via hydrolysis and settling.  ()
where fopp = the fraction of the phytoplankton internal phosphorus that is in organic form which is calculated as  ()
The fraction of the bottom algae internal phosphorus that is in organic form, fopb, is calculated in a similar fashion. The rate of organic phosphorus hydrolysis is computed as  ()
where khp(T) = the temperature-dependent organic phosphorus hydrolysis rate [/d]. Organic phosphorus settling is determined as  ()
where vop = organic phosphorus settling velocity [m/d].
Inorganic phosphorus increases due to organic phosphorus hydrolysis and plant excretion. It is lost via plant uptake. In addition, a settling loss is included for cases in which inorganic phosphorus is lost due to sorption onto settleable particulate matter such as iron oxyhydroxides:  ()
where  ()
where vip = inorganic phosphorus settling velocity [m/d].
Inorganic suspended solids are lost via settling, Smi = – InorgSettl where  ()
where vi = inorganic suspended solids settling velocity [m/d].
Dissolved oxygen increases due to plant photosynthesis. It is lost via fast CBOD oxidation, nitrification and plant respiration. Depending on whether the water is undersaturated or oversaturated it is gained or lost via reaeration,  ()
where  ()
where ka(T) = the temperature-dependent oxygen reaeration coefficient [/d], os(T, elev) = the saturation concentration of oxygen [mgO2/L] at temperature, T, and elevation above sea level, elev.
The following equation is used to represent the dependence of oxygen saturation on temperature (APHA 1995)  ()
where os(T, 0) = the saturation concentration of dissolved oxygen in freshwater at 1 atm [mgO2/L] and Ta = absolute temperature [K] where Ta = T +273.15. The effect of elevation is accounted for by  ()
The reaeration coefficient (at 20 oC) can be prescribed on the Reach Worksheet. If reaeration is not prescribed (that is, it is blank or zero for a particular reach), it is computed as a function of the river’s hydraulics and (optionally) wind velocity,  ()
where kah(20) = the reaeration rate at 20 oC computed based on the river’s hydraulic characteristics [d?1], KL(20) = the reaeration mass-transfer coefficient based on wind velocity [m/d], and H = mean depth [m]. Hydraulic-based Formulas: O’Connor-Dobbins (O’Connor and Dobbins 1958):  ()
where U = mean water velocity [m/s] and H = mean water depth [m]. Churchill (Churchill et al. 1962):  ()
Owens-Gibbs (Owens et al. 1964):  ()
Tsivoglou-Neal (Tsivoglou and Neal 1976): Low flow, Q = 0.0283 to 0.4247 cms (1 to 15 cfs):  ()
High flow, Q = 0.4247 to 84.938 cms (15 to 3000 cfs):  ()
where S = channel slope [m/m]. Thackston-Dawson (Thackston and Dawson 2001):  ()
where U* = shear velocity [m/s], and F = the Froude number [dimensionless]. The shear velocity and the Froude number are defined as  ()
and  ()
where g = gravitational acceleration (= 9.81 m/s2), Rh = hydraulic radius [m], S = channel slope [m/m], and Hd = the hydraulic depth [m]. The hydraulic depth is defined as  ()
where Bt = the top width of the channel [m]. USGS (Pool-riffle) (Melching and Flores 1999): Low flow, Q < 0.556 cms (< 19.64 cfs):  ()
High flow, Q > 0.556 cms (> 19.64 cfs):  ()
where Q = flow (cms). USGS (Channel-control) (Melching and Flores 1999): Low flow, Q < 0.556 cms (< 19.64 cfs):  ()
High flow, Q > 0.556 cms (> 19.64 cfs):  ()
Internal (Covar 1976): Reaeration can also be internally calculated based on the following scheme patterned after a plot developed by Covar (1976) (Figure ):
This is referred to as option Internal on the Rates Worksheet of Q2K. Note that if no option is specified, the Internal option is the default. 
Figure Reaeration rate (/d) versus depth and velocity (Covar 1976). Wind-based Formulas: Three options are available to incorporate wind effects into the reaeration rate: (1) it can be omitted, (2) the Banks-Herrera formula and (3) the Wanninkhof formula. Banks-Herrera formula (Banks 1975, Banks and Herrera 1977):  ()
where Uw,10 = wind speed measured 10 meters above the water surface [m s?1] Wanninkhof formula (Wanninkhof 1991):  ()
Note that the model uses Eq. (48) to correct the wind velocity entered on the Meteorology Worksheet (7 meters above the surface) so that it is scaled to the 10-m height.
Oxygen transfer in streams is influenced by the presence of control structures such as weirs, dams, locks, and waterfalls (Figure ). Butts and Evans (1983) have reviewed efforts to characterize this transfer and have suggested the following formula,  ()
where rd = the ratio of the deficit above and below the dam, Hd = the difference in water elevation [m] as calculated with Eq. (7), T = water temperature (?C) and ad and bd are coefficients that correct for water-quality and dam-type. Values of ad and bd are summarized in Table . If no values are specified, Q2K uses the following default values for these coefficients: ad = 1.25 and bd = 0.9. 
Figure Water flowing over a river control structure. Table Coefficient values used to predict the effect of dams on stream reaeration.
The oxygen mass balance for the element below the structure is written as  ()
where o’i?1 = the oxygen concentration entering the element [mgO2/L], where  ()
Pathogens are subject to death and settling,  ()
Pathogen death is due to natural die-off and light (Chapra 1997). The death of pathogens in the absence of light is modeled as a first-order temperature-dependent decay and the death rate due to light is based on the Beer-Lambert law,  ()
where kdX(T) = temperature-dependent pathogen die-off rate [/d] and ?path = a light efficiency factor [dimensionless].
Pathogen settling is represented as  ()
where vX = pathogen settling velocity [m/d].
The following equilibrium, mass balance and electroneutrality equations define a freshwater dominated by inorganic carbon (Stumm and Morgan 1996),  ()
 ()
 ()
 ()
 ()
where K1, K2 and Kw are acidity constants, Alk = alkalinity [eq L?1], H2CO3* = the sum of dissolved carbon dioxide and carbonic acid, HCO3? = bicarbonate ion, CO32? = carbonate ion, H+ = hydronium ion, OH? = hydroxyl ion, and cT = total inorganic carbon concentration [mole L?1]. The brackets [ ] designate molar concentrations. Note that the alkalinity is expressed in units of eq/L for the internal calculations. For input and output, it is expressed as mgCaCO3/L. The two units are related by  ()
The equilibrium constants are corrected for temperature by Harned and Hamer (1933):  ()
Plummer and Busenberg (1982):  ()
Plummer and Busenberg (1982):  ()
The nonlinear system of five simultaneous equations (177 through 181) can be solved numerically for the five unknowns: [H2CO3*], [HCO3?], [CO32?], [OH?], and {H+}. An efficient solution method can be derived by combining Eqs. (177), (178) and (180) to define the quantities (Stumm and Morgan 1996)  ()
 ()
 ()
where ?0, ?1, and ?2 = the fraction of total inorganic carbon in carbon dioxide, bicarbonate, and carbonate, respectively. Equations (179), (187), and (188) can be substituted into Eq. (181) to yield,  ()
Thus, solving for pH reduces to determining the root, {H+}, of  ()
where pH is then calculated with  ()
The root of Eq. (190) is determined with a numerical method. The user can choose either bisection, Newton-Raphson or Brent’s method (Chapra and Canale 2006, Chapra 2007) as specified on the QUAL2K sheet. The Newton-Raphson is the fastest but can sometimes diverge. In contrast, the bisection method is slower, but more reliable. Because it balances speed with reliability, Brent’s method is the default.
Total inorganic carbon concentration increases due to fast carbon oxidation and plant respiration. It is lost via plant photosynthesis. Depending on whether the water is undersaturated or oversaturated with CO2, it is gained or lost via reaeration,  ()
where  ()
where kac(T) = the temperature-dependent carbon dioxide reaeration coefficient [/d], and [CO2]s = the saturation concentration of carbon dioxide [mole/L]. The stoichiometric coefficients are computed as4  ()
 ()
The CO2 saturation is computed with Henry’s law,  ()
where KH = Henry's constant [mole (L atm)?1] and pCO2 = the partial pressure of carbon dioxide in the atmosphere [atm]. Note that the partial pressure is input on the Rates Worksheet in units of ppm. The program internally converts ppm to atm using the conversion: 10?6 atm/ppm. The value of KH can be computed as a function of temperature by (Edmond and Gieskes 1970)  ()
The partial pressure of CO2 in the atmosphere has been increasing, largely due to the combustion of fossil fuels (Figure ). Values in 2007 are approximately 10?3.416 atm (= 383.7 ppm). 
Figure Concentration of carbon dioxide in the atmosphere as recorded at Mauna Loa Observatory, Hawaii.5
The CO2 reaeration coefficient can be computed from the oxygen reaeration rate by  ()
As was the case for dissolved oxygen, carbon dioxide gas transfer in streams can be influenced by the presence of control structures. Q2K assumes that carbon dioxide behaves similarly to dissolved oxygen (recall Sec. 5.5.19.3). Thus, the inorganic carbon mass balance for the element immediately downstream of the structure is written as  ()
where c'T,i?1 = the concentration of inorganic carbon entering the element [mgO2/L], where  ()
where rd is calculated with Eq. (171).
As summarized in the present model accounts for changes in alkalinity due to several mechanisms: Table Processes that effect alkalinity.
According to Eq. (60), nitrification utilizes ammonium and creates nitrate. Hence, because a positive ion is taken up and a negative ion is created, the alkalinity is decreased by two equivalents. The change in alkalinity can be related to the nitrification rate by  ()
According to Eq. (61), denitrification utilizes nitrate and creates nitrogen gas. Hence, because a negative ion is taken up and a neutral compound is created, the alkalinity is increased by one equivalent. The change in alkalinity can be related to the denitrification rate by  ()
where the r’s are ratios that translate the processes into the corresponding amount of alkalinity. The stoichiometric coefficients are derived from Eqs. (58) through (61) as in
Hydrolysis of organic P results in the creation of inorganic phosphate. Depending on the pH, the phosphate will either have 1 (pH ? 2 to 7) or 2 (pH ? 7 to 12) negative charges. Hence, because negative ions are being created, the alkalinity is decreased by one or two equivalents, respectively. The change in alkalinity can be related to the P hydrolysis rate by6,  ()
where  ()
 ()
 ()
where Kp1 = 10–2.15, Kp2 = 10–7.2, and Kp3 = 10–12.35.
Hydrolysis of organic N results in the creation of ammonia. Depending on the pH, the ammonia will either be in the form of ammonium ion with a single positive charge (pH < 9) or neutral ammonia gas (pH > 9). Hence, when the positive ions are created, the alkalinity is increased by one equivalent. The change in alkalinity can be related to the N hydrolysis rate by  ()
Phytoplankton photosynthesis takes up nitrogen as either ammonia or nitrate and phosphorus as inorganic phosphate. If ammonia is the primary nitrogen source, this leads to a decrease in alkalinity because the uptake of the positively charged ammonium ions is much greater than the uptake of the negatively charged phosphate ions (recall the stoichiometry as described on p. 36). If nitrate is the primary nitrogen source, this leads to an increase in alkalinity because both nitrate and phosphate are negatively charged. The following representation relates the change in alkalinity to phytoplankton photosynthesis depending on the nutrient sources as well as their speciation as governed by the pH,  ()
Phytoplankton take up nitrogen as either ammonia or nitrate and phosphorus as inorganic phosphate. The following representation relates the change in alkalinity to phytoplankton uptake rates depending on the nutrient sources as well as their speciation as governed by the pH,  ()
Phytoplankton excrete ammonia and inorganic phosphate. The following representation relates the change in alkalinity to phytoplankton excretion rates including the effect of pH on the nutrient’s speciation,  ()
Bottom algae take up nitrogen as either ammonia or nitrate and phosphorus as inorganic phosphate. The following representation relates the change in alkalinity to bottom algae uptake rates depending on the nutrient sources as well as their speciation as governed by the pH,  ()
Bottom algae excrete ammonia and inorganic phosphate. The following representation relates the change in alkalinity to bottom algae excretion rates including the effect of pH on the nutrient’s speciation,  ()
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